Spontaneous self-assembling, such as formation of molecular crystals, is a fascinating topic for investigation. Ability to initiate and control such transformations promises numerous benefits, but our knowledge of underlying mechanisms of such processes is rather limited. The process of freezing of water is an excellent testing ground for such studies. In this paper we report the results of a systematic molecular dynamics study of ice growth at three different temperatures below the melting point initiated from a number of initial interface structures within the isoconfigurational ensemble. It is shown that a specific structure at a growing ice-water interface is able to affect the growth process over a time scale of 1-2 ns. This structural effect can be characterized in terms of relative growth propensities. On the basis of the differences in the shape between isoconfigurational rate distributions and the rate distribution typical of the specific temperature several different kinds of relative growth propensities have been identified. The initial interfacial configurations employed in this work have been assigned using the proposed classification and possible mechanisms of propensity realization have been suggested for selected cases. Results reported in this paper clearly indicate that local structure effects can have significant impact on tendency for a particular ice surface to grow (or melt). The structural effect on ordering propensities is, most probably, a more universal behaviour and might be expected to be seen in other similar problems such as, for example, protein folding.